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Abstract
We prove that for any given c, 1 < c < 17/11, almost all natural numbers are representable in the form [x c] + [p c], where x is a natural number and p is a prime.
Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E 3(N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P 5. Then we have E 3(N)≪log1053 N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P 11 in place of P 5.
Abstract
We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as
Abstract
Abstract
Let p
i
be prime numbers. In this paper, it is proved that for any integer k≧5, with at most 
exceptions, all positive even integers up to N can be expressed in the form 
. This improves the result 
for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1–4] for the problem in the form 
. This method can also be used for some other similar forms.
Abstract
Abstract
We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)−c(log log log log log n)) for some absolute c > 0.
Abstract
We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q 1 = Q 2:= n 1/2(log n)−ϑ and Q 3:= (log n) θ . Then for all q 3 ≦ Q 3, all reduced residues a 3 mod q 3, almost all q 2 ≦ Q 2, all admissible residues a 2 mod q 2, almost all q 1 ≦ Q 1 and all admissible residues a 1 mod q 1, there exists a representation n = p 1 + p 2 + p 3 with primes p i ≡ a i (q i ), i = 1, 2, 3.
Abstract
Let 
denote the set {n∣2|n, 
∀ p>2 with p−1|k}. We prove that when 
, almost all integers 
can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when 
, almost all integers n∊(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.
Abstract
It is proved that every sufficiently large even integer is a sum of one prime, one square of prime, two cubes of primes and 161 powers of 2.
